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Theorem mpt2xopxnop0 7573
Description: If the first argument of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, is not an ordered pair, then the value of the operation is the empty set. (Contributed by Alexander van der Vekens, 10-Oct-2017.)
Hypothesis
Ref Expression
mpt2xopn0yelv.f 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st𝑥) ↦ 𝐶)
Assertion
Ref Expression
mpt2xopxnop0 𝑉 ∈ (V × V) → (𝑉𝐹𝐾) = ∅)
Distinct variable groups:   𝑥,𝑦   𝑥,𝐾   𝑥,𝑉   𝑥,𝐹
Allowed substitution hints:   𝐶(𝑥,𝑦)   𝐹(𝑦)   𝐾(𝑦)   𝑉(𝑦)

Proof of Theorem mpt2xopxnop0
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 neq0 4128 . . 3 (¬ (𝑉𝐹𝐾) = ∅ ↔ ∃𝑥 𝑥 ∈ (𝑉𝐹𝐾))
2 mpt2xopn0yelv.f . . . . . . 7 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st𝑥) ↦ 𝐶)
32dmmpt2ssx 7465 . . . . . 6 dom 𝐹 𝑥 ∈ V ({𝑥} × (1st𝑥))
4 elfvdm 6437 . . . . . . 7 (𝑥 ∈ (𝐹‘⟨𝑉, 𝐾⟩) → ⟨𝑉, 𝐾⟩ ∈ dom 𝐹)
5 df-ov 6874 . . . . . . 7 (𝑉𝐹𝐾) = (𝐹‘⟨𝑉, 𝐾⟩)
64, 5eleq2s 2902 . . . . . 6 (𝑥 ∈ (𝑉𝐹𝐾) → ⟨𝑉, 𝐾⟩ ∈ dom 𝐹)
73, 6sseldi 3793 . . . . 5 (𝑥 ∈ (𝑉𝐹𝐾) → ⟨𝑉, 𝐾⟩ ∈ 𝑥 ∈ V ({𝑥} × (1st𝑥)))
8 fveq2 6405 . . . . . . 7 (𝑥 = 𝑉 → (1st𝑥) = (1st𝑉))
98opeliunxp2 5459 . . . . . 6 (⟨𝑉, 𝐾⟩ ∈ 𝑥 ∈ V ({𝑥} × (1st𝑥)) ↔ (𝑉 ∈ V ∧ 𝐾 ∈ (1st𝑉)))
10 eluni 4629 . . . . . . . . 9 (𝐾 dom {𝑉} ↔ ∃𝑛(𝐾𝑛𝑛 ∈ dom {𝑉}))
11 ne0i 4119 . . . . . . . . . . . . 13 (𝑛 ∈ dom {𝑉} → dom {𝑉} ≠ ∅)
1211ad2antlr 709 . . . . . . . . . . . 12 (((𝐾𝑛𝑛 ∈ dom {𝑉}) ∧ 𝑉 ∈ V) → dom {𝑉} ≠ ∅)
13 dmsnn0 5808 . . . . . . . . . . . 12 (𝑉 ∈ (V × V) ↔ dom {𝑉} ≠ ∅)
1412, 13sylibr 225 . . . . . . . . . . 11 (((𝐾𝑛𝑛 ∈ dom {𝑉}) ∧ 𝑉 ∈ V) → 𝑉 ∈ (V × V))
1514ex 399 . . . . . . . . . 10 ((𝐾𝑛𝑛 ∈ dom {𝑉}) → (𝑉 ∈ V → 𝑉 ∈ (V × V)))
1615exlimiv 2023 . . . . . . . . 9 (∃𝑛(𝐾𝑛𝑛 ∈ dom {𝑉}) → (𝑉 ∈ V → 𝑉 ∈ (V × V)))
1710, 16sylbi 208 . . . . . . . 8 (𝐾 dom {𝑉} → (𝑉 ∈ V → 𝑉 ∈ (V × V)))
18 1stval 7397 . . . . . . . 8 (1st𝑉) = dom {𝑉}
1917, 18eleq2s 2902 . . . . . . 7 (𝐾 ∈ (1st𝑉) → (𝑉 ∈ V → 𝑉 ∈ (V × V)))
2019impcom 396 . . . . . 6 ((𝑉 ∈ V ∧ 𝐾 ∈ (1st𝑉)) → 𝑉 ∈ (V × V))
219, 20sylbi 208 . . . . 5 (⟨𝑉, 𝐾⟩ ∈ 𝑥 ∈ V ({𝑥} × (1st𝑥)) → 𝑉 ∈ (V × V))
227, 21syl 17 . . . 4 (𝑥 ∈ (𝑉𝐹𝐾) → 𝑉 ∈ (V × V))
2322exlimiv 2023 . . 3 (∃𝑥 𝑥 ∈ (𝑉𝐹𝐾) → 𝑉 ∈ (V × V))
241, 23sylbi 208 . 2 (¬ (𝑉𝐹𝐾) = ∅ → 𝑉 ∈ (V × V))
2524con1i 146 1 𝑉 ∈ (V × V) → (𝑉𝐹𝐾) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1637  wex 1859  wcel 2158  wne 2977  Vcvv 3390  c0 4113  {csn 4367  cop 4373   cuni 4626   ciun 4708   × cxp 5306  dom cdm 5308  cfv 6098  (class class class)co 6871  cmpt2 6873  1st c1st 7393
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1880  ax-4 1897  ax-5 2004  ax-6 2070  ax-7 2106  ax-8 2160  ax-9 2167  ax-10 2187  ax-11 2203  ax-12 2216  ax-13 2422  ax-ext 2784  ax-sep 4971  ax-nul 4980  ax-pow 5032  ax-pr 5093  ax-un 7176
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1865  df-sb 2063  df-eu 2636  df-mo 2637  df-clab 2792  df-cleq 2798  df-clel 2801  df-nfc 2936  df-ne 2978  df-ral 3100  df-rex 3101  df-rab 3104  df-v 3392  df-sbc 3631  df-csb 3726  df-dif 3769  df-un 3771  df-in 3773  df-ss 3780  df-nul 4114  df-if 4277  df-sn 4368  df-pr 4370  df-op 4374  df-uni 4627  df-iun 4710  df-br 4841  df-opab 4903  df-mpt 4920  df-id 5216  df-xp 5314  df-rel 5315  df-cnv 5316  df-co 5317  df-dm 5318  df-rn 5319  df-res 5320  df-ima 5321  df-iota 6061  df-fun 6100  df-fv 6106  df-ov 6874  df-oprab 6875  df-mpt2 6876  df-1st 7395  df-2nd 7396
This theorem is referenced by:  mpt2xopx0ov0  7574  mpt2xopxprcov0  7575
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